Notes on Algebraic Cycles and Homotopy theory

TL;DR

This paper proves a conjecture by Lawson regarding the connectivity of Chow varieties of algebraic cycles, and computes their homotopy and homology groups up to a certain degree, extending results to arbitrary characteristic fields.

Contribution

It establishes the 2d-connectedness of Chow varieties of algebraic p-cycles and computes their homotopy groups, including étale homotopy groups over fields of any characteristic.

Findings

Confirmed Lawson's conjecture on Chow variety connectivity

Calculated homotopy and homology groups up to degree 2d

Extended results to algebraically closed fields of arbitrary characteristic

Abstract

We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}(P^n)$ of all effective algebraic p-cycles of degree d in n-dimensional projective space to the space of effective algebraic p-cycles is 2d-connected. As a result, the homotopy and homology groups of $C_{p,d}(P^n)$ are calculated up to 2d. We also show an analogous statement for Chow variety $C_{p,d}(\P^n)$ over algebraically closed fields of arbitrary characteristic and compute their etale homotopy groups up to 2d.